Conditional probabilities – what if…

August 25th, 2008, 2:02pm by Sam Wang

Today, reader Steve asked about the overall odds if Obama or McCain wins CO, OH, or VA. This is easy to calculate – I did it back in 2004 (scroll way down to October 22). So let’s revisit that…

Update: additions in italics are meant to clarify an issue. By the way, that paragraph about the Meta-Margin is important, and addresses issues of true prediction.

If the probabilities of a McCain win in these three states are p1, p2, and p3, then the probability of Obama winning one or more of the three is 1 – (p1 * p2 * p3), which works out to about 89% as of today.

How is McCain’s overall snapshot win probability affected by a certain win in any one of those states? After running EV_estimator.m, geeks may run the following code:

These 3 lines give the conditional probability of a McCain win if voters followed today’s polls in all states except for one forced win. (Conditional probability is a central concept in probability and statistics! If you feel a moment of confusion, read this.) Under the various forced-win scenarios, the conditional probabilities are:

In other words, if McCain loses even one of these three, it’s bleak for him. If he wins all three, his overall snapshot win probability is 36%, basically a toss-up. Also, the importance of Colorado means that Denver’s a good place for the Democrats to have their convention.

However, these forced-win scenarios are not the best way to think about a future scenario in which McCain wins. Depending on current state polls, which are the best predictor of an election held today, McCain’s win probability in one or more of those states could be quite small at any given moment. As a result, an ad hoc, forced-win case (a.k.a. conditional probability) is not the right question. Instead, a better way to think about this question is that to have a chance, McCain’s overall support relative to Obama has to move over the next two months by a certain amount – the Popular Meta-Margin, which is given at the top of this page. Today the Meta-Margin is fairly small, about 3%.

This is just one example of what’s possible with the Meta-Analysis. Many more features are possible.

Update: Once again people are interpreting the result here as a win probability of >99% for Obama. Sigh. No. The reason the probability is high is that the measurement is precise. However, November is a long way off. This is why you should be looking at the Meta-Margin.